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MathGroup Archive 2006

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Sequences from chemical structures as Hadamard self-similar matrices

  • To: mathgroup at smc.vnet.net
  • Subject: [mg68646] Sequences from chemical structures as Hadamard self-similar matrices
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Mon, 14 Aug 2006 06:44:25 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

My work on group theory representations taught me
how to repartition arrays in Mathematica
so that I could do Hadamard self-similarity
in fairly large matrices.
Although they aren't techically a new kind of fractal,
no one before has been able to express chemical sturctures as matrices 
like this
as far as I know.

I thought of how to do it last night
and got it working for 2by2's and 3by3's this morning.
This 6by6 gives a crystal like matrix structure for Trigonal prisms. ( 
triangles like that in a spectral prism)
These matrices appear to be both fractal and tile-like!

Clear[M, v, a, aaa, t]
t[n_, m_] := {{0, 1, 1, 1, 0, 0},
     {1, 0, 1, 0, 1, 0},
     {1, 1, 0, 0, 0, 1},
     {1, 0, 0, 0, 1, 1},
     {0, 1, 0, 1, 0, 1},
     {0, 0, 1, 1, 1, 0}}[[n, m]]
a = Table[t[n, m]*t[i, j], {n, 1, 6}, {m, 1, 6}, {i, 1, 6}, {j, 1, 6}]
M = Flatten[Table[{Flatten[Table[a[[
     n, m]][[1, i]], {n,
          1, 6}, {i, 1, 6}]], Flatten[Table[a[[n, m]][[2,
           i]], {n, 1, 6}, {i, 1, 6}]], Flatten[Table[a[[
     n, m]][[3, i]], {n,
          1, 6}, {i, 1, 6}]], Flatten[Table[a[[n, m]][[4,
           i]], {n, 1, 6}, {i, 1, 6}]], Flatten[Table[a[[
     n, m]][[5, i]], {n,
          1, 6}, {i, 1, 6}]], Flatten[Table[a[[n, m]][[6,
           i]], {n, 1, 6}, {i, 1, 6}]]}, {m, 1, 6}], 1]
ListDensityPlot[M, Mesh -> False]
v[1] = Table[Fibonacci[n], {n, 0, 35}]
v[n_] := v[n] = M.v[n - 1]
a = Table[v[n][[1]], {n, 1, 50}]
Det[M - x*IdentityMatrix[36]]
Factor[%]
aaa = Table[x /. NSolve[Det[M - x*IdentityMatrix[36]] == 0, x][[n]], {n, 1,
           36}]
Abs[aaa]
a1 = Table[N[a[[n]]/a[[n - 1]]], {n, 7, 50}]


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